Florida B.E.S.T. MA.912.C.4.7

MathGrades 9–12Develop an understanding for and determine integrals.

The Standard

Evaluate definite and indefinite integrals by using integration by substitution.

Florida B.E.S.T. Standards for Mathematics

Teacher's field guide

What Students Need to Do: Students identify a composite expression to use as u and rewrite the integral in terms of u and du. They evaluate the simpler integral, then substitute back or change the bounds. They distinguish when to include +C.
What Mastery Looks Like: Students choose a substitution that turns the integrand into a simpler form, then integrate accurately. They include +C for indefinite integrals and correctly handle bounds for definite integrals. They can check an indefinite result by differentiating it.
Common Misconceptions: Students choose u but cannot match du to the remaining factor. They forget the constant of integration, change bounds incorrectly, or mix x and u in the same integral.
How to Assess It: Use a two-part exit ticket: Evaluate ∫2x cos(x^2) dx and ∫ from 0 to 1 of 2x(x^2+3)^2 dx, showing the substitution.

Lesson moves

Give pairs cards showing integrals, possible substitutions, differentials, and antiderivatives, then have them build complete matching sets.
Ask students to explain why u = x^2 works for ∫2x cos(x^2) dx and why u = 2x does not help.
Play substitution bingo by displaying integrals while students cover the matching choice of u and transformed differential on their boards.
Use the rate R(t) = 2t(t^2 + 1)^3 liters per minute to find water added from t = 0 to 2.

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